# Using Dimensional Regularization to Tame Divergent Momentum Integrals

• kassem84
In summary, dimensional regularization is a useful technique for addressing divergence in momentum integrals. In a calculation involving three integrals over k_1, k_2, and k_3, the integral over k_3 may result in an ultraviolet divergence while the others give finite numbers. It is possible to manipulate these integrals before applying dimensional regularization, but caution must be taken if performing a multiloop calculation with minimal subtraction. In this case, all integrals should be dimensionally regularized with the same parameter and the d->4 limit should only be taken at the end. Any differences can be corrected using finite counterterms and a defined renormalization scheme.
kassem84
Dear all,
Dimensional regularization is a very important technique to remove the divergence from momentum integrals.
Suppose that you have to calculate a quantity composed of three integrals over k_1, k_2 and k_3 (each one is three dimensional). the integral over k_3 gives ultra violet divergence. Whereas, the remained integrals give finite numbers.
I have some questions:
1) Can we play with these integrals (performing change of variables or calculating one or two of these integrals) before performing dimensional regularization?
2) Can we transform these 3 integrals into one dimensional integrals that is divergent and then do the dimensional regularization?

Thanks in advance.
Best regards.

If you are only doing a one-loop calculation then you can be really slack and do all those things you mentioned. If you are doing a multiloop calculation and using minimal subtraction then you have to be more careful and have everything dimensionally regularized (with the same dimension parameter) throughout the whole calculation, only taking the d->4 limit at the end.
Of course, the only difference should be in finite counterterms that can be fixed using well defined renormalization scheme.

Thanks very much, it is a useful note.

## 1. What is dimensional regularization?

Dimensional regularization is a mathematical technique used to handle divergent integrals in quantum field theory. It involves analytically continuing the number of dimensions in a space-time from 4 to a complex number, and then taking the limit as this number approaches 4.

## 2. Why is dimensional regularization used in quantum field theory?

Dimensional regularization is used in quantum field theory because it provides a consistent method for dealing with divergent integrals that arise in calculations. It also preserves important symmetries and avoids the need for arbitrary cutoffs or renormalization schemes.

## 3. How does dimensional regularization differ from other regularization methods?

Unlike other regularization methods, which introduce a cutoff or make use of a finite number of dimensions, dimensional regularization allows for the use of infinite dimensions and uses analytic continuation to manipulate the integrals. This makes it a more elegant and powerful technique for handling divergent integrals.

## 4. What are the advantages of using dimensional regularization?

Some advantages of dimensional regularization include its ability to preserve important symmetries, its consistency in calculations, and its avoidance of arbitrary cutoffs or renormalization schemes. It also allows for a more systematic approach to dealing with divergent integrals, making calculations more manageable and less prone to errors.

## 5. Are there any limitations or drawbacks to using dimensional regularization?

While dimensional regularization is a powerful and widely used technique, it does have some limitations. It may not work in certain cases where there are non-analytic dependencies on the number of dimensions, and it may not be applicable to all types of divergent integrals. Additionally, it can be a more complex and challenging method to apply compared to other regularization methods.

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